I am trying to assign variable to register here is the code:
ORG 100h
var1 DB 10 ;
var2 DB 20 ;
MOV BX,var1 ; error : oper
You're using an assembler that keeps track of how you declare symbols to figure out what operand size to use.
var1 isn't an 8 bit address, it's a 16bit address (not counting the segment) that points at the first of two 8-bit variables. So the assembler error message is badly worded and confusing.
NASM would do what you said, and do a 16bit load. You'd find var1 in bl and var2 in bh. Presumably you could write mov bx, word [var1], or word ptr or whatever, to get your assembler to emit the 16bit load.
(Actually, NASM would assemble mov BX, var1 into a mov r16, imm16, putting the address into the register. Always use [] around memory references, for consistency, because that works in NASM and MASM variants of Intel syntax. NASM doesn't support the mov BX, offset var1 syntax for writing the mov-immediate form, though.)